Which of the following statements is/are true?
(A) $(\tan^{-1}y-x)dy = (1 + y^2)dx$ is a differential equation where variables are separable.
(B) $(1+x^2)dy+ 2xydx = \cot x dx (x ≠ 0)$ is a first order linear differential equation.
(C) $(4x+6y+5)dy - (3y+2x+4)dx = 0$ is not a homogeneous differential equation.
(D) $(xy)dx-(x + y^2)dy = 0$ is a homogeneous differential equation.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (C) only
(A) False. Rewrite as $\frac{dy}{dx}=\frac{1+y^{2}}{\tan^{-1}y-x}$; variables are not separable (equation is linear in $x$ when viewed as $dx/dy$).
(B) True. Rewrite as $\frac{dy}{dx}+\frac{2x}{1+x^{2}}\,y=\frac{\cot x}{1+x^{2}}$, a first-order linear ODE in $y$ (for $x\ne0$).
(C) True. $ (4x+6y+5)\,dy-(3y+2x+4)\,dx=0$ contains nonzero constant terms (5,4), so the functions are not homogeneous of the same degree; hence not homogeneous.
(D) False. $M(x,y)=xy$ is homogeneous of degree 2 but $N(x,y)=-(x+y^{2})$ is not homogeneous of the same degree, so the equation is not homogeneous.
True statements: (B) and (C)